Related papers: Holonomic approximation through convex integration
In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property…
In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity…
This paper consists of four general parts: convex sets; convex functions; convex optimization; and the interior-point algorithm. I will start by introducing the definition of convex sets and give three common convex set examples which will…
A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically…
In this paper, we propose a systematic approach for extending first-order optimization algorithms, originally designed for unconstrained strongly convex problems, to handle closed and convex set constraints. We show that the resulting…
Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the…
We give a geometrical demonstration to the existence of holomorphic first integrals for certain kind of vector fields in $\mathbb{C}^2$ and $\mathbb{C}^3$.
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of…
In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method,…
In this master's thesis, we introduce expansion systems as a general framework to describe a large variety of approximation algorithms, such as Taylor approximation, decimal expansion and continued fraction. We consider some basic…
The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and…
This paper is an attempt to remedy the problem of slow convergence for first-order numerical algorithms by proposing an adaptive conditioning heuristic. First, we propose a parallelizable numerical algorithm that is capable of solving…
Plane arrangements are a useful tool for surface and volume modelling. However, their main drawback is poor scalability. We introduce two key novelties that enable the construction of plane arrangements for complex objects and entire…
It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree…
In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a…
In this paper we propose an algorithm for exact partitioning of high-order models. We define a general class of $m$-degree Homogeneous Polynomial Models, which subsumes several examples motivated from prior literature. Exact partitioning…
In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth…
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of…